Polynomial Division. You may also see this kind of problem written like this: Perform the division x2 +2x 3

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Bryan Logan
5 years ago
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1 Polynomial Division 5015 You do polynomial division the way you do long division of numbers It s difficult to describe the general procedure in words, so I ll work through some eamples stepbystep Eample Find the quotient and remainder when + 3 is divided by You may also see this kind of problem written like this: Perform the division + 3 Set up the division the way you d set up division of numbers To start, look at the first term () in and the first term ( ) in Ask yourself: What times gives? You can see that works, so put it on top: (?) + 3 Net, multiply the by the on top, and put the result under + 3 Line up terms with the same power of : Subtract: ( +) ( ) (Whenyou subtract, be careful of the signs! Inthis case, the terms cancel, but ( ) + 4) Net, bring down the 3 and put it net to the 4: Look at the first term () in and the first term (4) in

2 Ask yourself: What times gives 4? (?) 4 You can see that 4 works, so put it on top: Multiply the by the 4 on top, and put the result 4 8 under 4 3 Line up terms with the same power of : (You don t multiply by the +4 on top, just the 4; you already multiplied by the in +4 in an earlier step) Subtract: (4 3) (4 8) At this point, the of doesn t go into 5, so the division is finished The quotient is +4, the epression on the top And the remainder is 5 If you re just asked for the quotient and remainder, you re done If the problem asked you to do the division + 3, then you d write it this way: (+4)+ 5 The quotient +4 goes in front The remainder 5 goes on top of the fraction The epression, which was on the bottom of the original fraction, goes on the bottom of the new fraction If you have trouble remembering where everything goes, think about how you convert improper fractions to mied numbers Let s say you want to convert 38 by : Then to a mied number To do this, you divide 38

3 However, if you think about it, 5 3 which you read as 5 and 3 means 5+ 3, so This is the same as what I did with the polynomials: The quotient 5 goes in front The remainder 3 goes on top of the fraction The epression, which was on the bottom of the original fraction 38, goes on the bottom of the new fraction In other words, as often happens when you re doing algebra, you can figure out what to do with variables by thinking about what you know to do with numbers Eample Find the quotient and remainder when 3 +1 is divided by +1 Since 3 +1 is missing an term, I ll write 0 as a placeholder It is optional, but this makes it less likely that you ll make a mistake when you do the subtraction Look at the first term () in +1 and the first term ( 3 ) in 3 +1: Ask yourself: What times gives 3? You can see that works, so put it on top: (?) Net, multiply +1 by the on top, and put the result 3 + under 3 +1 Line up terms with the same power of : Note that the goes under the 0 I put in as a placeholder Subtract: ( 3 +0 ) ( 3 + ) Net, bring down the and put it net to the :

4 Look at the first term () in +1 and the first term ( ) in : Ask yourself: What times gives? You can see that works, so put it on top: (?) Multiply +1 by the on top (just the you multiplied by the ), and put the result under : Subtract: ( ) ( ) Bring down the 1 since 0+1 1, I just write the 1: Since +1 does not go into 1, the division is finished:

5 The quotient is and the remainder is 1 In fraction form, this would be written as ( ) Eample Find the quotient and remainder when is divided by +1 Look at the first term ( ) in +1 and the first term (3 3 ) in : Ask yourself: What times gives 3 3? You can see that 3 works, so put it on top: (?) Net, multiply +1 by the 3 on top, and put the result under Line up terms with the same power of : Subtract: (3 3 +1) (3 6 +3) Net, bring down the 1 and put it net to the 5 10: Look at the first term ( ) in +1 and the first term (5 ) in 5 10:

6 Ask yourself: What times gives 5? You can see that 5 works, so put it on top: (?) Multiply +1 by the 5 on top (just the 5 you already multiplied by the 3 ), and put the result under : Subtract: (5 10+1) (5 10+5) Since +1 does not go into, the division is finished The quotient is 3+5 and the remainder is In fraction form, this would be written as Eample , or ( 3)( 1) 6 Eample , or ( 5)(+4) Sine the remainder is 0, this shows that factors as (+4)( 5) 6

7 The Remainder Theorem If p() is a polynomial, the remainder when p() is divided by a is p(a) Eample Divide p() by : ( )( )+4 The remainder is 4 On the other hand, p() The remainder is the same as p() The Root Theorem a divides p() evenly if and only if p(a) 0 (ie a is a root of p()) Eample Let p() p( 1) Therefore, 1 should divide In fact, ( 1)( ) Eample Let p() 6+8 Then p() ( )( 4) Since and 4 are factors, and 4 should be roots and they are: p() , p(4) Eample If you know or can guess a factor, you can sometimes complete a factorization by long division For eample 3 has 3 as a root By the fact I stated earlier, this means that 3 is a factor Divide 3 into 3 to get +3+9 Therefore, 3 ( 3)( +3+9) c 01 by Bruce Ikenaga

SECTION 2.3: LONG AND SYNTHETIC POLYNOMIAL DIVISION

2.25 SECTION 2.3: LONG AND SYNTHETIC POLYNOMIAL DIVISION PART A: LONG DIVISION Ancient Example with Integers 2 4 9 8 1 In general: dividend, f divisor, d We can say: 9 4 = 2 + 1 4 By multiplying both sides